#512227

order	512
nilpotency class	4
isoclinism	isoclinic class of #511747
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

499 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,4,5)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,28)(19,30,24,25)(20,33,21,26)(23,38,37,43)(27,35,46,36)(29,32,47,40)(31,41,34,44)(39,45,48,42);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,22,24)(18,25,28,30)(20,34,21,31)(23,32,37,40)(26,44,33,41)(27,42,46,45)(29,43,47,38)(35,48,36,39);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,20,22,21)(18,26,28,33)(19,31,24,34)(23,35,37,36)(25,41,30,44)(27,43,46,38)(29,45,47,42)(32,39,40,48);(17,21,22,20)(18,27,28,46)(19,31,24,34)(23,39,37,48)(25,42,30,45)(26,43,33,38)(29,44,47,41)(32,35,40,36)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gh,h^a=hi,c^b=ci,g^d=gh,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419423);